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Properties

Label 1-4640-4640.3579-r0-0-0
Degree $1$
Conductor $4640$
Sign $-0.388 - 0.921i$
Analytic cond. $21.5480$
Root an. cond. $21.5480$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)3-s − i·7-s − i·9-s + (−0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s − i·17-s + (−0.707 − 0.707i)19-s + (0.707 − 0.707i)21-s − i·23-s + (0.707 − 0.707i)27-s − i·31-s + 33-s + (−0.707 + 0.707i)37-s − 39-s − 41-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s − i·7-s − i·9-s + (−0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s − i·17-s + (−0.707 − 0.707i)19-s + (0.707 − 0.707i)21-s − i·23-s + (0.707 − 0.707i)27-s − i·31-s + 33-s + (−0.707 + 0.707i)37-s − 39-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4640\)    =    \(2^{5} \cdot 5 \cdot 29\)
Sign: $-0.388 - 0.921i$
Analytic conductor: \(21.5480\)
Root analytic conductor: \(21.5480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4640} (3579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4640,\ (0:\ ),\ -0.388 - 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3152355387 - 0.4749107308i\)
\(L(\frac12)\) \(\approx\) \(0.3152355387 - 0.4749107308i\)
\(L(1)\) \(\approx\) \(0.7197881016 - 0.08415448264i\)
\(L(1)\) \(\approx\) \(0.7197881016 - 0.08415448264i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 - iT \)
31 \( 1 - iT \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 - T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + (0.707 - 0.707i)T \)
71 \( 1 - iT \)
73 \( 1 + T \)
79 \( 1 + iT \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 - T \)
97 \( 1 + iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.162126351364122049627851007, −17.8036392126931893035717471494, −16.71914827596421693960630207052, −16.5667046958062385458100331965, −15.87730426814480382451014486682, −15.26470661261984433324952760201, −14.257427824536260156911147364987, −13.773443433195925334104697560196, −13.10994716058669978847259073038, −12.141854741601560384648541133545, −11.51408545638313625179625542774, −10.80942494684313208727783026964, −10.4573896076238625583791066437, −9.66193364235636071638102695222, −8.924891046635474503406978446690, −8.17034353334235813986364220613, −7.19809349672688886089762834402, −6.6452193960032048273557105246, −5.7714075744790040917443009488, −5.17469393052386278951320386019, −4.35421430787450246415611535895, −3.68446229902641669385060309887, −3.10275299446593427449885733649, −1.74425672367448551330533594376, −0.83682021396770121404652738305, 0.211737092887316947325592528657, 1.41884368684854798941922486635, 2.23192490791368389231929295850, 2.76526761080456353573303185711, 4.00091547621619515774801593594, 4.92931451419234313324102891364, 5.460414995917171104674779229535, 6.27871864622457900496006677155, 6.65741005149744163320183361762, 7.78241936536847522851300702942, 8.22092893998200404220193329370, 8.92433307915850273192141174797, 9.96437985064509171158743729143, 10.73591888627778954913708791506, 11.082745168216130719890018811586, 12.175702301337501997036690528034, 12.50447474718577944890125427751, 13.07349132627952207413044386128, 13.72634049748666955525825339006, 14.80641818342604023435942175569, 15.43059694942876047576223656275, 15.80636359922018123282467279465, 16.913808334400598117015743728503, 17.33935107756210218463602440366, 18.02407622210614300185881532320

Graph of the $Z$-function along the critical line